Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (3^1/2)(3^3/2)

Verified step by step guidance

1. Identify the expression to simplify: \((3^{1/2})(3^{3/2})\).
2. Recall the property of exponents that states when multiplying powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
3. Add the exponents: \(\frac{1}{2} + \frac{3}{2} = \frac{4}{2}\).
4. Simplify the sum of the exponents: \(\frac{4}{2} = 2\).
5. Rewrite the expression with the new exponent: \$3^2$, which is the simplified form without negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. For multiplication with the same base, add the exponents: a^m * a^n = a^(m+n). This rule allows combining terms like (3^(1/2)) and (3^(3/2)) by adding their exponents.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, a^(1/2) means the square root of a, and a^(3/2) means a^(1/2) cubed. Understanding fractional exponents helps in interpreting and simplifying expressions involving roots.

Eliminating Negative Exponents

Negative exponents indicate reciprocals, such as a^(-n) = 1/a^n. The problem requires answers without negative exponents, so any negative powers must be rewritten as positive exponents in the denominator to maintain clarity and standard form.